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Conjectures connecting number theory and geometry
In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry.
Langlands_program
Mathematical theory
the geometric Langlands correspondence relates algebraic geometry and representation theory. It is a reformulation of the Langlands correspondence obtained
Geometric Langlands correspondence
Geometric_Langlands_correspondence
Canadian mathematician
Robert Phelan Langlands (/ˈlæŋləndz/; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web
Robert_Langlands
Theorem in abstract algebra
[clarification needed] It was conjectured by Robert Langlands (1983) in the course of developing the Langlands program. The fundamental lemma was proved by Gérard
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Equivalence of two physical theories
Montonen–Olive duality is closely related to a research program in mathematics called the geometric Langlands program. Another realization of S-duality in quantum
S-duality
Mathematical concept
equivalence between motivic and automorphic L-functions postulated in the Langlands program can be tested. Automorphic forms realized in the cohomology of a Shimura
Shimura_variety
Russian-American mathematician
Frenkel introduced the analytic Langlands correspondence, a novel function-theoretic framework for the Langlands Program in the case of Riemann surfaces
Edward_Frenkel
Mathematics award
major recent progress on the geometric Langlands program, including the final proof of the geometric Langlands conjecture in characteristic zero." 2026
Breakthrough Prize in Mathematics
Breakthrough_Prize_in_Mathematics
Mathematical object
the Weil group. It was named after Robert Langlands by Robert Kottwitz. In Kottwitz's formulation, the Langlands group should be an extension of the Weil
Langlands_group
French mathematician (1952–2025)
2025) was a French mathematician working in number theory and the Langlands program. Laumon was born in 1952. He studied at the École Normale Supérieure
Gérard_Laumon
French mathematician
mathematician who is active in algebraic geometry, especially in the Langlands program, and a CNRS "Directeur de Recherches" at the Institute Fourier in
Vincent_Lafforgue
Malaysian mathematician (born 1972)
context of the Langlands program, especially the theory of theta correspondence, the Gan–Gross–Prasad conjecture and the Langlands program for Brylinski–Deligne
Gan_Wee_Teck
French mathematician
outstanding contributions to Langlands' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism
Laurent_Lafforgue
Chinese mathematician (born 1982)
primarily with geometric representation theory and in particular the Langlands program, tying number theory to algebraic geometry and quantum physics. Zhu
Xinwen_Zhu
British mathematician who proved Fermat's Last Theorem
Theorem, but pushed the whole of mathematics as a field towards the Langlands program of unifying number theory. Wiles was born on 11 April 1953 in Cambridge
Andrew_Wiles
\varepsilon (\pi \otimes \chi ,1/2)} is the Langlands ε {\displaystyle \varepsilon } -constant [ (Langlands 1970); (Deligne 1972) ] associated to π {\displaystyle
Waldspurger_formula
Branch of algebraic number theory concerned with abelian extensions
absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence
Class_field_theory
Romania mathematician
London. Her research interests include algebraic number theory and the Langlands program. She was born in Bucharest and studied at Mihai Viteazul High School
Ana_Caraiani
Israeli-American mathematician
Mathematics (MPIM) at Bonn and is known for his research on the geometric Langlands program. Born in Chișinău (now in Moldova) he grew up in Tajikistan, before
Dennis_Gaitsgory
View of mathematicians to consolidate two or more theories into a more generalized one
should be fitted into one theory (examples include Hilbert's program and Langlands program). The unification of mathematical topics has been called mathematical
Unifying theories in mathematics
Unifying_theories_in_mathematics
Mathematical conjectures in class field theory
In mathematics, the local Langlands conjectures, introduced by Robert Langlands, are part of the Langlands program. They describe a correspondence between
Local_Langlands_conjectures
Branch of mathematics that studies abstract algebraic structures
invariant theory and the Erlangen program, has an impact in number theory via automorphic forms and the Langlands program. There are many approaches to representation
Representation_theory
Completes the Langlands program for general linear groups over algebraic function fields
groups and representations of Galois groups. The Langlands conjectures were introduced by Langlands (1967, 1970) and describe a correspondence between
Lafforgue's_theorem
Postgraduate center in New Jersey, US
and the IAS maintains the key repository for the papers of Langlands and the Langlands program. The IAS is a main center of research for homotopy type theory
Institute_for_Advanced_Study
Conjecture in the representation theory of Lie groups
_{\mathrm {GP} }} be the "distinguished character" defined in terms of the Langlands–Deligne local constant, then furthermore Hom H ( π ( φ , η ) ⊗ ν ¯
Gan–Gross–Prasad_conjecture
Mathematical group representations with same data
have the same Langlands parameter, and so have the same L-function and ε-factors. L-packets were introduced by Robert Langlands in (Langlands 1989), (Labesse
L-packet
Type of Dirichlet series associated to number field extensions
larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm
Artin_L-function
British mathematician (born 1980)
mathematician working in number theory and arithmetic aspects of the Langlands Program. He specialises in algebraic number theory. Gee was awarded the Whitehead
Toby_Gee
Low-rank isomorphisms in mathematics
are used in the formulation of Langlands functoriality, local and global transfer, and instances of the local Langlands correspondence. A related application
Exceptional isomorphisms of classical groups
Exceptional_isomorphisms_of_classical_groups
In mathematics, representation of a reductive algebraic group
f\left({\begin{pmatrix}1&b\\0&1\end{pmatrix}}g\right)=\tau (b)f(g).} Jacquet & Langlands (1970) used Whittaker models to assign L-functions to admissible representations
Whittaker_model
Mathematics independent of applications
theory and authors like Robert Langlands advocate for the unification of mathematics with Physics through the Langlands program. Mathematicians have always
Pure_mathematics
Gives conditions for the solvability of quadratic equations modulo prime numbers
vast generalization of quadratic reciprocity. Robert Langlands formulated the Langlands program, which gives a conjectural vast generalization of class
Quadratic_reciprocity
It has been one of the most powerful techniques for studying the Langlands program. The theory in some sense dates back to Bernhard Riemann, who constructed
Rankin–Selberg_method
Pro-algebraic group
In mathematics, the Serre group S is the pro-algebraic group whose representations correspond to CM-motives over the algebraic closure of the rationals
Serre_group
Austrian mathematician (born 1948)
Laumon, G.; Rapoport, M.; Stuhler, U. (1993). "D-elliptic sheaves and the Langlands correspondence". Inventiones Mathematicae. 113 (1). Springer Science and
Michael_Rapoport
In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair
Theta_correspondence
German mathematician
algebraic groups over the p-adic numbers, with connections to the Langlands program. She is a professor at the University of Bonn. Fintzen competed for
Jessica_Fintzen
Mathematical group
endoscopic groups of reductive algebraic groups were introduced by Robert Langlands (1979, 1983) in his work on the stable trace formula. Roughly speaking
Endoscopic_group
In number theory, the Shimura correspondence is a correspondence between modular forms F of half integral weight k+1/2, and modular forms f of even weight
Shimura_correspondence
Relates rational elliptic curves to modular forms
is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation
Modularity_theorem
American-Israeli mathematician
2013 "for contributions to the development and dissemination of the Langlands program." Gelbart was born in Syracuse, New York, son of the mathematician
Stephen_Gelbart
Russian American mathematician (born 1957)
representations of quantum groups and Hecke algebras, and on the geometric Langlands program (Satake equivalence of categories). He is currently a Professor of
Victor_Ginzburg
36 mathematical problems stated in 1955
Theorem in 1995. Taniyama's problems influenced the development of the Langlands program, the theory of modular forms, and the study of elliptic curves. Taniyama's
Taniyama's_problems
Proposition in mathematics that is unproven
that the first conjecture is true and the second one is false. The Langlands program is a far-reaching web of these ideas of 'unifying conjectures' that
Conjecture
explain critical aspects of consciousness? Geometric Langlands and physics: can the Langlands program related to representation theory explain the symmetries
List of unsolved problems in physics
List_of_unsolved_problems_in_physics
Mathematics award
Chicago, US "Drinfeld's main preoccupation in the last decade [are] Langlands' program and quantum groups. In both domains, Drinfeld's work constituted a
Fields_Medal
Korean educator (born 1978)
California, Berkeley working in number theory, automorphic forms, and the Langlands program. From 1994 to 1996 when he was in Seoul Science High School, Shin
Sug_Woo_Shin
Topics referred to by the same term
Langlands dual Langlands group Langlands program Langlands, Queensland, a locality in the Western Downs Region, Queensland, Australia Langlands Park, a rugby
Langlands
French mathematician (1928–2014)
"visionary program". The ℓ-adic cohomology then became a fundamental tool for number theorists, with applications to the Langlands program. Grothendieck's
Alexander_Grothendieck
Russian mathematician (born 1962)
framework for a Langlands program for higher-dimensional schemes, and with, Victor Ginzburg and Éric Vasserot, extended the "Geometric Langlands Conjecture"
Mikhail_Kapranov
special case of Langlands functoriality, corresponding (roughly) to the diagonal embedding of the Langlands dual GLn(C) of GLn to the Langlands dual GLn(C)×
Base_change_lifting
American-Canadian mathematician (born 1941)
He is closely connected to the Langlands program and has been involved in posting all of the work of Robert Langlands on the internet. Casselman did his
Bill_Casselman
1970 mathematics text by Jacquet and Landlands
Godement, R. (1970), Notes on Jacquet–Langlands' theory, Institute for Advanced Study Jacquet, H; Langlands, R. P. (1970), Automorphic Forms on GL (2)
Automorphic_Forms_on_GL(2)
Mathematical concept
their functional equation being first proved via the Langlands–Shahidi method. In general, the Langlands functoriality conjectures imply that automorphic
Automorphic_L-function
Branch of pure mathematics
area of research in algebraic number theory is Iwasawa theory. The Langlands program, one of the main current large-scale research plans in mathematics
Number_theory
Israeli mathematician
in Paris.[citation needed] Braverman specializes in the geometric Langlands program, the intersection of number theory, algebraic geometry and representation
Alexander_Braverman
Hungarian mathematician
manifolds, Yang–Mills instantons, non-Abelian Hodge theory, Geometric Langlands program, and representation theory of quivers and Kac–Moody algebras. Hausel
Tamás_Hausel
historical and recent, include Felix Klein's Erlangen program, Hilbert's problems, Langlands program, and the Millennium Prize Problems. In the Mathematics
Future_of_mathematics
American theoretical physicist
Edward (April 21, 2006). "Electric-Magnetic Duality And The Geometric Langlands Program". Communications in Number Theory and Physics. 1: 1–236. arXiv:hep-th/0604151
Edward_Witten
Chinese-American mathematician (born 1982)
geometry and representation theory, with a particular focus on the Langlands program. He was previously a C. L. E. Moore instructor at Massachusetts Institute
Zhiwei_Yun
American mathematician
verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many
Thomas_Callister_Hales
Study of algorithms for performing number theoretic computations
conjecture, the Sato-Tate conjecture, and explicit aspects of the Langlands program. Magma computer algebra system SageMath Number Theory Library PARI/GP
Computational_number_theory
American mathematician (born 1972)
He works in algebraic number theory and arithmetic aspects of the Langlands program. Skinner was born on June 4, 1972, in Little Rock, Arkansas. Skinner
Christopher_Skinner
French mathematician
Waldspurger (born 2 July 1953) is a French mathematician working on the Langlands program and related areas. He proved Waldspurger's theorem, the Waldspurger
Jean-Loup_Waldspurger
enunciated by Israel Gelfand, and the philosophy is a precursor of the Langlands program. A consequence for thinking about representation theory is that cuspidal
Parabolic_induction
German mathematician
his contributions to the Langlands program. In 1993, he—along with Gérard Laumon and Michael Rapoport—proved the local Langlands conjectures for the general
Ulrich_Stuhler
K-types were introduced by Vogan as part of an algebraic description of the Langlands classification. Vogan, David A. (January 1979). "The Algebraic Structure
Minimal_K-type
British mathematician
Imperial College London. He specialises in arithmetic geometry and the Langlands program. While attending the Royal Grammar School, High Wycombe he competed
Kevin_Buzzard
Galois group of the rationals by the Serre group. It was introduced by Langlands (1977) using an observation by Deligne, and named after Yutaka Taniyama
Taniyama_group
Robinson presents non-standard analysis. 1967 – Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation
Timeline_of_mathematics
Generalisation of a sheaf; a fibered category that admits effective descent
scheme is a stack. A moduli stack of shtukas is used in geometric Langlands program. (See also shtukas.) Constructing weighted projective spaces involves
Stack_(mathematics)
Australian-American mathematician
mathematics at the University of Chicago working in number theory and the Langlands program. Frank Calegari was born on December 15, 1975. He has both Australian
Frank_Calegari
Mathematics award
including the foundations of theoretical physics" "For his work on the Langlands program" 1999 Andrew Wiles "For his role in the development of number theory"
Clay_Research_Award
Analytic function on the upper half-plane with a certain behavior under the modular group
forms and elliptic curves. Robert Langlands built on this idea in the construction of his expansive Langlands program, which has become one of the most
Modular_form
German mathematician
mathematics of the Deutsche Forschungsgemeinschaft for her work on the Langlands program. She was an invited speaker at the 2018 International Congress of
Eva_Viehmann
On the distribution of prime numbers
relate it to automorphic forms and Langlands program. Despite it is no solution for the problem and Langlands program itself is still highly conjectural
Hilbert's_eighth_problem
leads to the Langlands program: if G {\displaystyle G} is a reductive algebraic group and P = M A N {\displaystyle P=MAN} is the Langlands decomposition
Langlands_decomposition
\left({\begin{pmatrix}a&b\\0&1\end{pmatrix}}\right)f(x)=\tau (bx)f(ax).} Jacquet & Langlands (1970) showed that an irreducible representation of dimension greater
Kirillov_model
Mathematic theory
Bernstein, Joseph; Gelbart, Stephen (eds.), An introduction to the Langlands program (Jerusalem, 2001), Boston, MA: Birkhäuser Boston, pp. 109–131,
Tate's_thesis
of Shimura varieties. Hilbert modular surface Siegel modular variety Langlands, Robert P.; Ramakrishnan, Dinakar, eds. (1992), The zeta functions of
Picard_modular_surface
American foundation
to October 4, 2024. Notable workshops include: New Advances in the Langlands Program: Geometry and Arithmetic New Frontiers in Probabilistic and Extremal
Clay_Mathematics_Institute
Area of mathematics
Arithmetic geometry Arithmetic dynamics Topological quantum field theory Langlands program Sikora, Adam S. "Analogies between group actions on 3-manifolds and
Arithmetic_topology
Mathematics prize
contributions to arithmetic geometry and number theory: in particular, the Langlands program.". List of mathematics awards "Prizes and Awards". American Mathematical
Ruth Lyttle Satter Prize in Mathematics
Ruth_Lyttle_Satter_Prize_in_Mathematics
System of partial differential equations used in Higgs field theory
used by Ngô Bảo Châu in his proof of the fundamental lemma in the Langlands program, for which he was afforded the 2010 Fields Medal. The definition may
Hitchin's_equations
Area of mathematical analysis
as a separate subject closely connected with number theory and the Langlands program. Convergence of Fourier series Fourier analysis for computing periodicity
Harmonic_analysis
Algebraic structure with addition, multiplication, and division
group are fundamental in many branches of arithmetic, such as the Langlands program. The cohomological study of such representations is done using Galois
Field_(mathematics)
Theory of subatomic structure
Witten, Edward (2007). "Electric-magnetic duality and the geometric Langlands program". Communications in Number Theory and Physics. 1 (1): 1–236. arXiv:hep-th/0604151
String_theory
French mathematician (born 1979)
the supervision of Gérard Laumon. Her thesis made progress on the Langlands program. After her Ph.D., she was a Clay Research Fellow between 2005 and
Sophie_Morel
Canadian mathematician
representation theory, p-adic groups, motivic integration, and the Langlands program. Gordon earned her PhD at the University of Michigan in 2003 under
Julia_Gordon
Conjecture on zeros of the zeta function
be equivalent, this is important open problem itself and part of Langlands program. Artin (1924) introduced global zeta functions of (quadratic) function
Riemann_hypothesis
Type of generalization of periodic functions in Euclidean space
generalizations with various algebro-geometric properties; and the resultant Langlands program. To oversimplify, automorphic forms in this general perspective, are
Automorphic_form
French mathematician
local Langlands conjecture, and at the same time introduces extra structure which mirrors the more categorical formulation of the geometric Langlands conjecture
Laurent_Fargues
Module over a sheaf of differential operators
G. D-modules are also crucial in the formulation of the geometric Langlands program. Hotta, Takeuchi & Tanisaki 2008, p. 18. Hotta, Takeuchi & Tanisaki
D-module
Mathematician
published a short article that expanded the scope of the Langlands conjectures. The Langlands conjectures, when published in 1967, could be seen as a sort
Vladimir_Drinfeld
Mathematical concept
and the Main Conjecture. The proof of the fundamental lemma in the Langlands program also made use of techniques that reduced the number field case to
Global_field
Geometric space whose points represent algebro-geometric objects of some fixed kind
Bundles" (PDF). Moduli theory Moduli stacks in P-adic modular forms and Langlands program Grothendieck, Alexander (1960–1961). "Techniques de construction en
Moduli_space
Australian mathematician
Using this theory, together with the work of Colmez on the p-adic Langlands program, he posted a preprint in 2011 proving many cases of the Fontaine--Mazur
Matthew_Emerton
Sheaf cohomology on the étale site
ISBN 978-3-540-57116-2 Archibald and Savitt Étale cohomology Goresky Langlands Program For Physicists Milne, James S. (1998), Lectures on Étale Cohomology
Étale_cohomology
Sporadic simple group
results "also represent the intersection of moonshine theory with the Langlands program, which, since its inception in the 1960s, has become a driving force
O'Nan_group
Mathematical theorem
ℓ. An alternative version of the reciprocity law, leading to the Langlands program, connects Artin L-functions associated to abelian extensions of a
Artin_reciprocity
LANGLANDS PROGRAM
LANGLANDS PROGRAM
Surname or Lastname
English and Scottish
English and Scottish : topographic name, from Old English lang, long ‘long’ + land ‘land’, ‘territory’.Norwegian : variant of Langeland.
Surname or Lastname
English (chiefly Lancashire)
English (chiefly Lancashire) : habitational name from Leyland in Lancashire (recorded in Domesday Book as Lailand), or from Laylands in Yorkshire; both are named from Old English lǣge ‘untilled ground’ + land ‘land’, ‘estate’. In some cases the name may be topographical.
Boy/Male
Arabic
Way; Program
Boy/Male
Muslim
Way. Program.
Boy/Male
Arabic, Muslim
Way; Program; Road; Path
Surname or Lastname
English
English : habitational name from any of the numerous places named with Old English lang ‘long’ + lÄ“ah ‘wood’, ‘glade’; or a topographic name with the same meaning.English : from the Old Norse female personal name LanglÃf, composed of the elements lang ‘long’ + lÃf ‘life’.English : Americanized spelling of French Langlais.
LANGLANDS PROGRAM
LANGLANDS PROGRAM
Boy/Male
Tamil
Shyamanga | à®·à¯à®¯à®¾à®®à®‚கா
Dark skinned one
Boy/Male
Indian, Punjabi, Sikh
The Sun
Girl/Female
Irish
Melancholy. Aolder name Deirdre. In Celtic legend Deirdre died of a broken heart.
Girl/Female
African, American, British, Christian, English, Gujarati, Hindu, Indian, Japanese, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Flower; Natural; Everywhere; God; Born During the Summer; Beautiful Line; To Ask; Good Mother
Girl/Female
American, Finnish, Hindu, Indian, Latin, Spanish, Swedish
Blessed; Good Person
Girl/Female
Hindu, Indian, Marathi, Tamil
One with Beautiful Hair
Girl/Female
Hebrew Spanish
Given by God.
Boy/Male
Hindu, Indian
One who is Happy in Following Dharma
Girl/Female
Australian, Christian, Czechoslovakian, Danish, Dutch, Finnish, French, German, Greek, Polish, Slovenia, Swedish
Peace
Female
Norwegian
 Danish and Norwegian variant spelling of Scandinavian Kristina, KRISTINE means "believer" or "follower of Christ." Compare with another form of Kristine.
LANGLANDS PROGRAM
LANGLANDS PROGRAM
LANGLANDS PROGRAM
LANGLANDS PROGRAM
LANGLANDS PROGRAM
n.
An elaborate instrumental composition for a full orchestra, consisting usually, like the sonata, of three or four contrasted yet inwardly related movements, as the allegro, the adagio, the minuet and trio, or scherzo, and the finale in quick time. The term has recently been applied to large orchestral works in freer form, with arguments or programmes to explain their meaning, such as the "symphonic poems" of Liszt. The term was formerly applied to any composition for an orchestra, as overtures, etc., and still earlier, to certain compositions partly vocal, partly instrumental.
n.
See Programme.
n.
Anything that is scattered abroad in great numbers as a theatrical programme, an advertising leaf, etc.
n.
A preface.
v. t.
A list of candidates, prepared for nomination or for election; a list of candidates, or a programme of action, devised beforehand.
n.
Same as Programme.
n.
An edict published for public information; an official bulletin; a public proclamation.
n.
Any law, which, after it had passed the Athenian senate, was fixed on a tablet for public inspection previously to its being proposed to the general assembly of the people.
pl.
of Programma
n.
A printed programme of a play, with the parts assigned to the actors.
n.
That which is written or printed as a public notice or advertisement; a scheme; a prospectus; especially, a brief outline or explanation of the order to be pursued, or the subjects embraced, in any public exercise, performance, or entertainment; a preliminary sketch.
n.
A published note, containing a brief statement, explanation, request, expression of thanks, or the like; as, to put a card in the newspapers. Also, a printed programme, and (fig.), an attraction or inducement; as, this will be a good card for the last day of the fair.